There are many definitions of 'god'. I'll start off by making it clear the definitions I'm not dealing with. I'm not considering 'gods' who are intelligent and powerful aliens who can do things that seem magical (a good example of such an alien is 'Q' in Star Trek). I'm not dealing with beings who can create a world that seems real to us, such as the Machines in the Matrix trilogy. I'm not dealing with the version of the Christian god written about by the physicist Frank Tipler who attempts to explain miracles in terms of physics in his book "The Physics of Christianity". Why aren't I dealing with such gods? Because they aren't what most believers want gods to be - they can't provide ultimate judgement and ultimate forgiveness; they can't give ultimate meaning; they can't provide eternal bliss or eternal punishment. What I'm dealing with is beings that have powers that are truly 'supernatural', and that includes the Christian god - the Alpha, the Omega, the creator of all things and the source of all morality.
I have a couple of arguments that deal with the question of evidence for such beings:
1. The argument from complexity.
The Catholic Church insists that their god is ultimate simplicity, but that's just not on. A being that is infinite, eternal and all-knowing and all-powerful is exceedingly - perhaps infinitely - complex, as that being contains all knowledge, and all wisdom. This complexity is a real problem when it comes to evidence for this god, as just about anything else is simpler. This includes vast galactic civilizations that have existed for billions of years. It includes Star Trek-level cultures that can destroy a world with a phaser bank, and can cure most illnesses with a wave of something that looks like a pepper pot with lights. So, if you come across what seems like a miracle, or you have some internal mental experience that feels like religious revelation, there are many alternatives of lesser complexity you have to consider before you allow for the possibility of the Catholic god. The complexity problem has been expressed beautifully by Arthur C. Clarke, who said 'any sufficiently advanced technology will be indistinguishable from magic', and by David Hume, who said that claims of miracles are never to be trusted, because there are always simpler explanations.
2. The argument from supernaturalness
The word 'supernatural' is the label for attributes of gods which are 'beyond Nature'. The problem with this label is that it's never specified what 'beyond Nature' is supposed to mean. Nature as we know it involves particles like atoms, electrons, photons and so on. So, presumably, a supernatural being manages to get things done in ways that don't involve any such particles. But that isn't an explanation of what they are actually doing to perform miracles. Even if you can have reliable evidence that what is happening doesn't involve familiar particles, that evidence is in no way evidence for 'beyond Nature', it's only 'beyond what we know'. So, from a practical point of view, evidence for the supernatural is definitely a problem. It gets worse when we consider that a common definition of supernatural is 'beyond the reach of science'. This makes evidence for the supernatural impossible by definition.
It's worth at this point clearing up a common misconception. Sometimes evidence is considered to be supportive of the supernatural, when what that evidence is actually for is a thing that is believed to be supernatural. For example, a primitive tribe might consider planes flying over their rain forest to be gods. When asked for evidence of these supernatural gods by another tribe, they point up at a metal machine high above. Of course, planes aren't supernatural (although I have to say that they feel like magic to me). What I mean by 'evidence for the supernatural' is evidence that a thing has supernatural nature.
So, whichever definition we choose for 'supernatural', we reach an impasse. We either have to try and demonstrate that something is beyond Nature, which is impossible, or we have a property of beings that is defined as being beyond empirical testing, so demonstrating its supernatural nature is impossible.
So, gods, by their definitions, are beyond reach of evidence. No evidence is sufficient to show that what seems like a god or an act of a god isn't some simpler alternative, and according to some definitions, evidence isn't even possible to test a god's divine supernatural nature.